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1 ASYNCHRONOUS MIPS PROCESSORS: EDUCATIONAL SIMULATIONS A Thesis Presented to the Faculty of California Polytechnic State University San Luis Obispo In Partial Fulfillment of the Requirements for the Degree Master of Science in Computer Science by Robert Webb July 2010

4 Abstract Asynchronous MIPS Processors: Educational Simulations Robert Webb The system clock has been omnipresent in most mainstream chip designs. While simplifying many design problems the clock has caused the problems of clock skew, high power consumption, electromagnetic interference, and worst-case performance. In recent years, as the timing constraints of synchronous designs have been squeezed ever tighter, the efficiencies of asynchronous designs have become more attractive. By removing the clock, these issues can be mitigated. However, asynchronous designs are generally more complex and difficult to debug. In this paper I discuss the advantages of asynchronous processors and the specifics of some asynchronous designs, outline the roadblocks to asynchronous processor design, and propose a series of asynchronous designs to be used by students in tandem with traditional synchronous designs when taking an undergraduate computer architecture course. iv

5 Acknowledgements Thank you to my parents for paying my tuition, among other things. v

11 Chapter 1 Introduction Computer chip designers have long relied on a system wide clock to signal the change of a state. This has resulted in designs where all instructions are assumed to introduce the worst possible delay. This bottleneck has been mitigated by solutions such as pipelining, branch prediction, and dynamic scheduling. However, the recent slowdown in the progress of designers to further increase the clock rates of chips has led to renewed interest in designing a processor without a clock. Using an internal clock requires that a chip can only progress as fast as the slowest component, which has created increasingly complex pipelines. Even with these very complex pipelines, much of the chip is inactive for a large portion of the time between cycles. Also, as clock rates increase, the size of the circuits has become an issue because of the need for the clock pulse to reach every component of the chip before the next pulse occurs. Asynchronous design is a possible solution to many of these roadblocks. This thesis presents the basic theory of asynchronous circuit design. To further demonstrate these ideas, simulations of asynchronous processors are introduced. The simulations are designed in parallel with popular synchronous designs to facilitate their use as pedagogical tools, as the lack of academic resources has been one of the principal reasons asynchronous designs have not reached more widespread awareness. 1

12 The available literature does not contain a small simulation of a working asynchronous processor with laboratory exercises. The primary contribution of this paper is to fill that gap. The simulations detailed here are small and easily accessible. The first could be introduced in a few additional lectures once a single cycle synchronous design has been discussed. Depending on the detail desired, introducing the pipelined design to a class could take anywhere from a few days to a few weeks of lecture and lab. 1.1 Advantages of Asynchronous Design The advantages of asynchronous designs can be categorized primarily in terms of speed increase, elimination of the clock skew problem, reduction of electromagnetic interference, and lower power consumption Potential Speed Increase A processor that does not have a clock can proceed when processing is complete. There is no need to wait for the next synchronizing clock cycle. For example, when using a simple cascading adder, the addition of two small numbers is fast. However, the subtraction of two small numbers takes quite a bit of time because the two s complement representation of negative numbers requires that all of the significant bits be determined in a cascade. The ALU pipeline stage must be long enough for the worst possible case, but an asynchronous processor can move on once the calculation is completed Elimination of Clock Skew As clock rates have become faster the speed of signals across the distance of the chip has become a problem. The maximum delay of the clock signal across the chip is the clock skew, and this delay must be small when compared 2

13 to the frequency of the clock. In modern designs methods of clock distribution have become increasingly complicated and expensive in order to mitigate this problem.[12] Creating processors without a clock eliminates this problem Electromagnetic Interference The clock pulse must be powerful enough to reach every part of the chip to synchronize operations. This powerful electric current traveling through the chip can cause significant electromagnetic interference for other nearby components. Eliminating the clock significantly reduces electromagnetic interference produced by processors because the total current will be lower and it will not pulse at a constant frequency.[12] Reduced Power Consumption Am important commercial interest in asynchronous processors has been their extremely low power consumption. The largest power requirement of a processor is the clock signal, which must occur even when the processor is idle [12]. An asynchronous processor never has to generate a clock signal, so uses less power when running. Also, upon a cache miss or other event requiring the processor to wait for other components of the system, the processor can maintain an idle state with essentially no power consumption. Because of these factors, an asynchronous design consumes significantly less power than synchronous processors. This makes an asynchronous processor an ideal candidate for a battery powered mobile device. 1.2 Roadblocks There are several roadblocks to producing asynchronous processors. These stem from the increased difficulty of asynchronous design and the inertia of industry and educational institutions in the direction of synchronous designs. Specif- 3

14 ically, there are few design tools, trained engineers, or academic courses on the subject Design Difficulties The lack of a synchronizing clock makes design much more difficult. Every part of the processor is acting independently, which can lead to unexpected interactions and behavior. Fixing the problems with one set of interacting systems can cause cascading problems down the pipeline Design Tools The few design tools that are available to engineers have not been widely adopted. Also, when compared to existing synchronous tools, the asynchronous design tools automate fewer parts of the design process and do not provide as much framework for debugging and testing. For more information on available tools see [1] Lack of Academic Courses Very few schools offer any kind of academic training in asynchronous design. This has led to most engineers being unaware of the theory that has been developed by research groups developing asynchronous designs. Because the theories have not been widely taught, asynchronous design problems can seem more daunting [12]. 4

15 1.3 Outline: Parallels to Computer Organization and Design The purpose of this paper is to propose a way of integrating asynchronous design into a traditional undergraduate computer architecture class using Patterson and Hennessy s Computer Organization and Design (COD) [11] as a starting point. The project began with a term paper at Cal Poly in Winter 2009 with Dr. John Seng. It was impractical to learn all the theory of asynchronous design in the timespan of the 10 week quarter, so the goal was to implement a basic single cycle asynchronous processor using mostly synchronous design principals from COD while borrowing asynchronous principals only when necessary. In the end, only dual rail encoding and basic dual rail gates (defined in section 3.1) were required. However, several timing assumptions were required to make the simulation simple enough to complete in the time allotted. The result was a design that was not robust. To expand the term project into a master s thesis, pipelining and the necessary theory to develop a robust design would be added. In order to integrate the work into existing courses, the design would continue to parallel COD. The final product would be a set of asynchronous simulations that could be used in a traditional course focused on synchronous designs to demonstrate asynchronous designs. To parallel the machine arithmetic chapter of COD, two asynchronous ALU units are presented. The first is naive, but easy to understand, and presented in section The second requires more theory, but is much more robust in that it has fewer timing assumptions, and the timing assumptions are mostly local in nature. The theory required to understand the second ALU is presented in section 4.1. The ALU design is presented in section To parallel the single cycle data-path chapter of COD, a simple asynchronous processor is presented. This design was created using the design from the text and converting the data path wires to dual-rail encoding and any logic involving 5

16 these wires to dual-rail gates. These busses are then used to create completion signals which take the place of the clock. As noted above, to make this simulation work, several naive assumptions needed to be made. This design is the subject of chapter 3. To parallel the pipelining chapter of COD, a pipelined processor using asynchronous design techniques is presented. This is presented in chapter 4. The chapter includes a detailed description of the theory that was incorporated into the design in section 4.1. The design of the simulation is detailed in section 4.2. This report is concluded with a description of the educational resources that accompany the project in chapter 5 and appendix D and a discussion of conclusions and possible future research in chapter 6. Also included as appendices are the complete source code for the single-cycle synchronous processor (appendix A), single-cycle asynchronous processor (appendix B), pipelined asynchronous processor (appendix C). 6

17 Chapter 2 Previous Work 2.1 Asynchronous Logic For small scale circuits the cumulative effects of gate and wire delay are not difficult to deal with. For this reason, many small circuits do not have a clock because one is simply not needed. The fact that these circuits were asynchronous is generally not explicitly stated. For a discussion of these circuits, see chapter 8 of [14]. When the number of switches or transistors in a circuit began to grow exponentially, quickly approaching the millions, the problem of synchronizing the behavior of a circuit became nontrivial. This was exacerbated by the problem that these pioneers were literally manufacturing the first computers, so all of these circuits had to be laid out by hand on paper. Because CAD simulations were not available, the solution that most designers used was to determine an upper bound on the worst case delay of a circuit or segment of a circuit and use an external clock to signal the completion of the logic and stability of the outputs. 7

18 2.2 Early Developments An exception to this trend toward synchronous designs were the designers of ILLIAC II at the University of Illinois in the late 1950 s and early 1960 s [10]. One of the first pipelined computers, it also had many components designed to work without an external clock. Much of the early theory of large scale asynchronous circuits was developed by the designers of ILLIAC II, specifically David E. Muller. Because, at the time, wire delay was insignificant compared to gate delay, a theory of Speed-Independent (SI) circuits was developed to generate logic that would not glitch given arbitrary gate delay. A SI circuit is assumed to have no wire delay. A new gate was developed to accomplish this, the Muller C-Element [10]. These developments are discussed in detail in section 4.1. The handshaking protocols used in the pipelined simulation were also developed in this period by Muller. This remained the standard handshaking protocol until 1988 when Ivan Sutherland proposed an improvement in [13], but this improvement is not used here because of the intended audience and additional circuit complexity. 2.3 Modern Developments The principal inspiration for this project has been the work of the Asynchronous Processor Group at the California Institute of Technology, led by Alain J. Martin. One of the early contributions of this research group was the paper that described the limitations of the successor to SI circuit theory, Delay-Insensitive (DI) circuit theory. As manufacturing processes improved to include more complex circuits with faster switching, it became clear that wire delay and gate delay were concurrent problems. This led to the development a theory of DI circuits where the delay of wires AND gates could be any arbitrary non-negative value without causing any unstable outputs. In the early 80 s, Martin developed a model for circuit timing similar to the context-free languages used in computability theory and language theory to prove that any circuit conforming to the requirements of DI would only 8

19 be functionally equivalent to a circuit made entirely from C-elements. As the C-element is not a universal gate 1, this creates a very limited set of potential DI circuits [7]. The limitations led to the development of a compromise to the strict requirements of DI. Called Quasi-Delay Insensitive (QDI), nearly all asynchronous development has followed this paradigm. The compromise is to simply allow a wire to fork out locally and guarantee that the data carried on that wire will reach all the local receivers at the same time. These are called iso-chronic forks. Manohar and Martin proved that these circuits are Turing-complete in [2] Because of the limitations of the original definition of DI and the adoption of QDI, most modern references to DI are, in fact, referring to the assumptions and requirements of QDI. In the 80 s, Cal Tech also developed the first QDI general purpose processor, mostly as a proof-of-concept. A RISC 2 design, it was finely pipelined and employed many opportunities for concurrent processing that would have been significantly more difficult in a synchronous design. Most significantly, the execute, memory, and write back stages of a traditional 5 stage pipeline were designed to run in parallel, asynchronously. Specifically, there were several busses to write back into the register file, and different instructions could simultaneously be accessing the ALU and memory modules, while asynchronously writing back to the registers. Results were promising, even with manufacturing defects. Most impressively, the processor was entirely QDI. This allowed it to run at a wide range of voltages and temperatures [3, 4]. The advancements in asynchronous design led to speculation that clockless processors were going to be much more widely used in the 90 s [5]. While this speculation proved incorrect, progress in asynchronous design continued. Martin published the design of a fast asynchronous adder in 1991 [6]. This design was optimized by going down to the transistor level, so was not used in this project as the intended audience for the simulation is computer science students, not 1 C-elements cannot be used to create a circuit that is equivalent to all the other standard logic gates. 2 Reduced Instruction Set Computing 9

20 electrical engineers, however, it could significantly increase the logic speed of asynchronous circuits. For a discussion of several asynchronous adders, including the one detailed in section and the one designed by Martin, see section 5.3 of [12]. As an additional proof-of-concept, Martin s group produced an asynchronous processor running the MIPS R3000 machine language [8, 9]. This processor is currently the state of the art in terms of asynchronous design and the next step in the logical development after [3]. The papers also developed a new metric for the performance of asynchronous processors, Eτ 2 where E is the average energy per instruction and τ is the average instruction execution time. Experiments suggested that, for a given design, that Eτ 2 is roughly constant because voltage varies inversely with the square of speed. 2.4 Educational Resources As asynchronous design has been the next big thing and just around the corner for quite some time, it has been only recently that the subject has been the subject of publications beyond short journal articles or technical reports. The lack of textbooks has been a roadblock to the teaching of asynchronous design because most available material was not very student-friendly. Two monographs have been written to remedy this. Davis and Nowick wrote a circuit design manual in 1997 is targeted more toward the experienced designer [1]. Sparsøwrote a textbook targeted to graduate students and advanced undergraduates in 2006 that is freely available on his web page [12]. This textbook has been invaluable to this project. It was the primary reference, especially for pipeline design, even though the principle pipeline control design dates back to Muller and Illiac II. As noted in section 1, these books do not include a small, student accessible, simulation of an asynchronous processor. 10

21 Chapter 3 A Single Cycle Asynchronous MIPS Processor This chapter describes a single cycle asynchronous MIPS processor that closely follows the designs presented in COD. To make the simulation accessible, asynchronous circuit theory is used as little as possible. The result is that the processor is simple, but not robust. 3.1 Design Principles Dual Rail Encoding In dual rail encoding every bit A is represented by a pair of bits AH and AL, for A-high and A-low. If A = 0 then AH = 0 and AL = 1. If A = 1 then AH = 1 and AL = 0. If A has not been fully determined, then AH = 0 and AL = 0. The situation where AH = AL = 1 has different meanings in each of the simulations. For the first simulation, this case will also imply that the bit is undetermined. For the more advanced pipeline simulation, the circuit will be designed so that this case cannot occur. See section for more on this more 11

22 Figure 3.1: basic dual rail NOT gate: NOT2 advanced paradigm Dual Rail Logic All the logic gates used in the traditional encoding of data have an equivalent gate that takes dual rail data as input and produces dual rail data as output. (AH, AL) = (AL, AH) so (1, 0) = (0, 1) and (0, 1) = (1, 0) as shown in figure 3.1. If A has not yet been determined then A will not be determined either (0, 0) = (0, 0). Similarly (AH, AL) (BH, BL) = (AH BH, AL BL) as shown in figure 3.2 and (AH, AL) (BH, BL) = (AH BH, AL BL) as shown in figure 3.3. These new gates are referred to as NOT2, AND2, and OR2 in the simulations using these constructions. As can be seen, if both inputs to either AND2 or OR2 are undetermined then the output is undetermined. If one of the inputs to AND2 is low then it will output low, even if the other is undetermined. If one of the inputs to AND2 is high then it will output undetermined if the other input is undetermined. If 12

23 Figure 3.2: basic dual rail AND gate: AND2 Figure 3.3: basic dual rail OR gate: OR2 one of the inputs to OR2 is high then it will output high, even if the other is undetermined. If one of the inputs to OR2 is low then it will output undetermined if the other input is undetermined. The fact that these new gates are actually more complicated circuits constructed from traditional gates implies that timing is still a concern. For example, if the AND gate inside the AND2 is slower than the OR gate inside AND2, then the output of the AND2 will produce an output of (1, 1) when transitioning from an output of (1, 0) to an output of (0, 1). This is an example of a glitch. Because of this problem an output of (1, 1) is defined as undetermined, and XOR is used to determine if the output of a circuit using these gates is ready to be used. While this method is used in the first simulation, a more sophisticated construction is used in the second simulation that will keep these glitches from occurring. This construction requires more theory to be developed which is detailed in

24 3.2 Simulation Because the target audience of this simuation is an undergraduate studying COD, a simple single stage abbreviated MIPS simulator has been altered to incorporate asynchronous design principles The Synchronous Starting Point The first design is a single state synchronous MIPS simulation capable of using the addi (add immediate), add (add register), sw (save word to memory), lw (load word from memory), j (jump), and bne (branch when not equal) instructions. A simple instruction memory, and data memory have been simulated using behavioral Verilog. 1 An ALU was constructed used structural Verilog code with the naive assumption that all AND, NOT, and OR gates generate one time unit of delay. The adder is a simple cascading design. A simple program has been written to test the functioning of the unit. 1 // Instruction Fetch Module 2 // Synchronous Version 3 // Robert Webb 4 5 module instructionfetch(pc, clk, instruction); 6 input [31:0] pc; 7 input clk; 8 output [31:0] instruction; 9 10 reg [31:0] instructions[0:56]; 11 reg [31:0] instruction; (posedge clk) #10 14 begin 15 if (pc > 60) $stop; 16 instruction = instructions[pc]; 17 end initial 20 begin 21 instructions[0] = 0; 22 instruction = instructions[0]; 1 Behavioral Verilog is a specification of the behavior of a circuit without specifying the physical design of the circuit. 14

25 23 instructions[4] = 32 b ; // addi $5, $0, 1 24 instructions[8] = 32 b ; // addi $4,$0,20 25 instructions[12] = 32 b ; // sw $0, $0(0) 26 instructions[16] = 32 b ; //addi $1, $0, 1 27 instructions[20] = 32 b ; //sw $1, $0(4) 28 //Loop 29 instructions[24] = 32 b ; //lw $6, $0(0) 30 instructions[28] = 32 b ; //lw $7, $0(4) 31 instructions[32] = 32 b ;//add $8, $7, $6 32 instructions[36] = 32 b ; //sw $7, $0(0) 33 instructions[40] = 32 b ; //sw $8, $0(4) 34 instructions[44] = 32 b ; //addi $4, $4, instructions[48] = 32 b ;// bne $4,$5,-7 36 //EndLoop 37 instructions[52] = 32 b ;// add $2,$8,$0 38 instructions[56] = 32 b ; // j instructions[60] = 32 b0; 40 end // initial begin 41 endmodule // instructionfetch At the conclusion of this program register $2 contains the 20 th Fibonacci number The remainder of the code for the traditional synchronous processor is presented as an appendix. With the previously stated assumptions, the above program will run with a clock frequency of 96 time units per cycle. At this speed the simulation completes in time units. These values are noted so they can be compared to the speed of the asynchronous designs Beginning the Asynchronous Conversion Verilog is a modular design tool, so the logical starting point is to define new modules corresponding to dual rail AND, OR, and NOT. Because NOT2 has no logical components, it was assumed to have no delay. For AND2 and OR2 to work properly, delay must be introduced so the undetermined bits will be logically obvious. Assuming that AND took one tick to transition up and two ticks to transition down while OR instantaneously transitioned up and took one tick to transition down produced the desired behavior. These delays were determined by running the simulation with different delays 15

26 while observing the outputs of the ALU. Using these delays, the undetermined bits cascaded down the outputs of the ALU as the carry rippled through the circuit, while guaranteeing that there was always an undetermined bit until all bits were defined. All the busses were then converted to dual rail encoding and all logical gates were converted to dual rail gates. 1 // Simple Dual-Rail Gates 2 // Asynchronous Processor 3 // Robert Webb 4 5 module not2(outh, outl, inh, inl); 6 input inh,inl; 7 output outh, outl; 8 9 buf (outh, inl), 10 (outl, inh); 11 endmodule // not module and2(outh, outl, AH, AL, BH, BL); 14 input AH,AL,BH,BL; 15 output outh, outl; and #(1,2) (outh, AH, BH); 18 or #(0,1) (outl, AL, BL); 19 endmodule // and module or2(outh, outl, AH, AL, BH, BL); 22 input AH,AL,BH,BL; 23 output outh, outl; or #(0,1) (outh, AH, BH); 26 and #(1,2) (outl, AL, BL); 27 endmodule // or Completion Signals The only parts of the processor that introduce significant and non-constant delay in this simulation are the 3 ALU units. The first ALU increments the current PC by 4. The second ALU increments the incremented PC by the signextended and shifted immediate from the current instruction. The completion signals from these units are combined to generate a signal referred to as PCready in the simulation specification. The final ALU performs the calculation required by the instruction. The completion signal from this unit is referred to as ALUready 16

27 in the simulation specification. To determine if an ALU has completed it s work, a module in Verilog is used. The basic circuit is shown in figure 3.4. Depending on the situation, all three may need to be finished to move on. If the instruction is not a branch, then the ALU that is computing branchpc is computing garbage, so the ALU adding the shifted, sign extended immediate and the incremented PC does not need to be finished. The simulated processor does take advantage of this fact in that the PCready signal is computed using the output of the multiplexor that selects nextpc. This allows PCready to signal without waiting for branchpc if it is not needed. The completion detector module takes a high and low output from an ALU or other processing unit 2 and returns a high bit if it has completed its work and a low bit if it has not. The determination is accomplished by computing the XOR of each high and low bits and then taking the AND of all of these (called good in the code). A BUF with a delay of 3 time units is then used to delay the output of the AND (called really in the code) which is then compared to the real-time output of the AND gate. The delayed signal is used to attenuate any glitches in the processor that might jump the gun. Attenuation is accomplished because BUF module in Verilog is internally defined to ignore any signals that do not remain high for the duration of the delay of the module. These signals are then passed through a behavioral bit of code that simply changes the possible bit value of x that is produced by Verilog at the beginning of the simulation into a 0. The behavioral code is only used to get the process started at the beginning because none of the processing units will initiate with an undefined completion signal. 1 // Completeness Indicator Module 2 // Asynchronous Processor 3 // Robert Webb 4 5 module ready(inh, inl, out); 6 input [31:0] inh, inl; 7 output out; 2 It is important that the output of a flip-flop never be used to generate a completion signal in the way described because a flip-flop could enter a metastable state. It is not needed here to generate a completion signal from the register file here, so this detail can be omitted, but in the case a completion signal is needed it can be produced by simply delaying the request signal going into the register file by an upper bound of latency of the flip-flops. 17

29 37 endmodule // ready Clock Removal In a single cycle implementation, the clock triggers 4 events: 1. Signal an update of the PC. 2. Signal a new instruction fetch. 3. Signal a write back of data to the register file. 4. Signal a write or read of the memory file. The goal of the simulation is to somehow generate a global completion signal (GCS) which will replace the clock. First, a distinction between the write or read function of the memory module is required. A read can happen when the ALU is finished because it is calculating the address of the read. This must occur before any GCS because the value of the read may be required by the register file, which will write back upon receiving the GCS. A write would occur upon receiving the GCS. Therefore, the memory module has been written to have distinct get and set signals. The get signal is connected to the ALUready signal. The set signal is connected to the GCS. 1 // Memory Module 2 // Asynchronous Processor 3 // Robert Webb 4 5 module memoryfile(readdatah, readdatal, addressh, addressl, 6 writedatah, writedatal, memwrite, memread, get, set); 7 input [31:0] addressh, addressl, writedatah, writedatal; 8 input get, set, memwrite, memread; 9 output [31:0] readdatah, readdatal; reg [31:0] readdatah = 0, readdatal = 0; 12 reg [31:0] sys[127:0]; (posedge get) 15 begin 16 if (memread) 19

30 17 begin 18 readdatah = sys[addressh]; readdatal = ~sys[addressh]; 19 $strobe("read %d from ad. %d %d \n", readdatah, addressh, $time); 20 end 21 end 22 (posedge set) 23 begin 24 if (memwrite) 25 begin 26 sys[addressh] = writedatah; 27 $strobe("wrote %d to ad. %d %d \n", writedatah, addressh, $time); 28 end 29 end 30 endmodule // memoryfile Distribution of the GCS To fully remove the clock, three ready modules for the PC, the main ALU, and the final output are used. These ready signals are combined with an AND gate to produce the GCS. The global ready signal is sent back into the PC, instruction fetch, register, and memory modules. Appropriate delays for the GCS going into each module are required to ensure correct operation. Specifically, the ready signal must first perform any writes to the memory or register units before the state changes. The latch holding the current PC can then be updated. The instruction fetch module must wait for a stable output from the latch to lookup the next instruction. A high level schematic of the processor, without control signals, is shown in figure 3.5. The processor is now asynchronous and independent of a clock signal. 3.3 Results and Discussion Design Lessons When this design was complete, one major problem was found. Branch instructions always caused the processor to go to a garbage address. The problem 20

32 was the original design simply checked if the main ALU was finished before signaling the next state change. After tracing through the simulation several times, it was determined that the branch address ALU calculation was not completed, so the simulation was jumping to a garbage address. Adding completion signals to all the ALUs fixed the problem, which occurs because it takes much longer for a cascade adder to subtract a small number than to add a small number, and the branch instruction is performing a subtraction to calculate branchpc. The bug is instructive of the design issues that asynchronous processor designers must grapple with that may not be as pervasive in a synchronous environment Results The same program that took the synchronous processor time units to complete was completed by the asynchronous version in 7604 time units. This is a 43.6% increase in computing speed. A significant amount of the gain is from the simple arithmetic done at the beginning of the Fibonacci calculation. Simple arithmetic strongly favors the asynchronous implementation because it can be completed by the ALU quickly. One would expect the speed gains to be less significant with more complex calculations that approached the worst case delay of a synchronous design. 22

33 Chapter 4 An Asynchronous Pipelined MIPS Processor 4.1 Theory As stated in chapter 2 the theory of asynchronous circuits begins with the Speed Independent (SI) model developed in the 50 s and 60 s by David Muller.[10] This theory assumes that all wires are ideal and have no delays and that gates can have arbitrary and potentially variable delay. A correctly designed SI circuit will never glitch, meaning that once the inputs are stable, the output bits will make a single transition to the correct output. An example of a glitch is described in section Theoretically correct asynchronous circuits will either produce a valid correct output, or an output that clearly communicates that it is not ready Delay Insensitivity Early computers did not run at a speed where wire delay would be an issue. As silicon design began to increase both clock speed and the number of gates that 23

34 could be used, wire delay became a significant part of the delay of the circuit. Because of this development, the logical next step was to develop a theory where wires and gates could have an arbitrary and potentially variable positive delay, known as DI. However, as was shown by Martin in [7], the class of circuits that are DI is very limited, so this theory was largely abandoned. When modern literature refers to DI, generally the assumptions are that of QDI Quasi-Delay Insensitivity The standard compromise to DI is Quasi-Delay Insensitivity (QDI). Using the QDI paradigm a designer can assume that a wire that forks will have the same delay in a local area. These are called iso-chronic forks. In other words, if a wire forks and goes to multiple nearby gates, a transition on that wire will reach the gates at the same time. It has been shown that these circuits are Turing complete.[2] From an electrical engineering perspective, QDI is a reasonable assumption that can be guaranteed with careful layout of a processor. However, complexity to the design process in increased because the forks that need to be iso-chronic have to be identified and monitored through fabrication to ensure correct operation. In the simulation, the forks required to be iso-chronic are noted The Muller C Element The basic building block of SI and QDI circuits is the Muller C element, first used in the ALU of Iliniac II. As noted in [12], the necessity of a new gate for asynchronous processors can be best described by the concept of indication. A gate is said to indicate an input if the value of the input can be completely determined by the output of the gate. Consider the two input AND gate. A transition of the output to high indicates that both inputs are high. However, a transition to low indicates that only one input is low, so no information can be determined about one of the inputs. Transitioning to high indicates the state of 24

35 both inputs, but transitioning to low indicates the state of only one. Similarly the OR gate also does not indicate both inputs on both transitions. For correct SI or QDI behavior, a gate that indicates both inputs on both transitions is required. It is useful to consider joining two request signals. If both are high, then the output should be high, and if both are low, then the output should be low. If only one is high, then the output should be low. A high joined signal means that that the processing for both is finished. A low combined signal means that both units are ready to do something else. 1 Therefore, if two requests are high and then one goes low, then their joined request should stay high because both of them are not ready to start processing again. The joined request should only go low when both of them are again low. This is the gate known as the Muller C element. It stays low until all inputs are high, then stays high until all inputs are low. The truth table is: A B C C t C t The gate description and transistor implementation are shown in figure 4.1. The code that has been written for 2, 3, and 32 input C elements follows this paragraph. Behavioral Verilog is used because these units are considered atomic. While they could be expressed using AND and OR gates, defining the C element with real AND and OR gates could introduce glitches because of different delays in AND and OR gates. 1 // C Elements 2 // Pipelined Asynchronous Processor 3 // Robert Webb 4 5 module Celement(a, b, c); 6 input a, b; 7 output c; reg c = 0; 11 always #1 if(a == 1 && b == 1) c = 1; 1 Request signals will be more completely discussed in section

37 Figure 4.2: C Element minterms for a 2 input DIMS circuit DIMS Once C elements have been defined it is possible to make circuits that, by definition, cannot glitch using a systematic algorithm. The algorithm is called Delay Insensitive Minterm Synthesis or DIMS. Using this method all of the data must be in dual-rail format as defined in section 3.1. The value of (1, 1) is not allowed in this case. The circuits will be designed so that that combination will never be produced as long as that combination is never an input and all transitions on inputs are to or from (0, 0). Specifically, an input of (1, 0) can never transition directly to (0, 1) or vise-versa. As an example, a simple 2 input circuit will be converted. 2 Because there are 4 possible input states, there will be 4 C elements in the circuit representing each possible minterm. 3 Figure 4.2 represents the first stage of the DIMS circuit. The truth table for the AND gate is: A B C To complete the DIMS algorithm the C element outputs associated with each minterm are connected to the appropriate output. If there are multiple minterms for a given output, they can be connected with an OR gate as each minterm is 2 Note that because these are dual-rail circuits, a 2 input circuit will have 4 input wires. 3 If not all inputs are possible, then some of the C elements can be removed. 27

39 Figure 4.4: DIMS OR gate 23 buf (outl, LL); 24 or #1 (outh, HH, LH, HL); 25 endmodule // andd The resulting circuits will satisfy all the requirements of the SI theory and the QDI theory. The only delay assumption is that the forks of the wires going into the C elements are isochronic. However, to use these circuits, the inputs must begin in the undefined state (i.e. all bits are at (0, 0)). By definition, in this state, the output will also be in the undefined state. The inputs can then change monotonically to their input state. In complex circuits with multiple outputs, the behavior of the outputs can be defined to be strongly indicating or weakly indicating. Strongly indicating circuits require all inputs to be defined before producing any output. Weakly indicating circuit outputs are allowed to become defined at different times and will produce output before all inputs are defined, if possible. For example, in the adder that is described in section 4.2.2, the single bit adders can produce output if the carry input is undefined because the carry out can be determined in some cases without the carry in value. Once the output is defined, care must be taken to be sure that all inputs return to the undefined state and that the inputs remain undefined until the 29

40 output has again returned to the undefined state. This is very important because of the hysteresis of the C elements. One possible consequence of not returning to the undefined state would be an output where both the high bit and low bit of an output is 1, which is not allowed in the circuit specification. The pipeline structure will ensure that the inputs and outputs return to the undefined state because the pipeline timing signals will be used to return the input to the undefined state and ensure the output has returned to the undefined state Design Choices Clearly the new AND and OR gates are much more complicated. A traditional AND gate has 6 transistors and the new one has 30 transistors. For this reason, the approach of making new AND and OR modules and changing all of the gates in a synchronous processor as was done in chapter 3 is unrealistic. Instead of doubling the transistor count as before, there would be a quintupling of the transistor count. To mitigate this, when converting a circuit the DIMS procedure is done on larger modules to reduce replication of overhead. However, this can quickly become daunting. Assuming there are no possible optimizations, an n input DIMS gate will have 2 n C elements each having n inputs! Modules have been created to convert between dual-rail data and a bundled data encoding. In dual-rail encoding of data, the validity of the data is encoded as part of the data. This works well when the data is being used by combinatorial logic and makes the calculation of a completion signal much easier, but is very cumbersome in other parts of a data path such as multiplexors and registers. For these parts of the data path bundled data is used. Bundled data is the traditional encoding of the data with an additional bit indicating if the data is valid. This additional bit is usually referred to as a request because if it is high, then the data is requesting to be consumed by some subsequent processing unit. The dual-rail to bundled data converter is labeled in schematics as SR, and the bundled data to dual -rail converter is labeled as DR. The circuit diagram for the DR converter is in figure 4.5. The circuit diagram for the 30